Title: Teachers beliefs and the reform movement in mathematics education. By: Battista, Michael T., Phi Delta Kappan, 00317217, Feb94, Vol. 75, Issue 6Database: Academic Search Premier
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TEACHER BELIEFS AND THE REFORM MOVEMENT IN MATHEMATICS EDUCATION
Through extensive education programs and institutional reform, we must help teachers become comfortable with the new view of mathematics, Mr. Battista says -- because, once they fully understand and believe in the reform movement, teachers will lead the way in implementing it.
Teachers are key to the success of the current reform movement in U.S. mathematics education. However, many teachers have beliefs about mathematics that are incompatible with those underlying the reform effort. Because these beliefs play a critical role not only in what teachers teach but in how they teach it, this incompatibility blocks reform and prolongs the use of a mathematics curriculum that is seriously damaging the mathematical health of our children. THE REFORM MOVEMENT
With its release of Curriculum and Evaluation Standards for School Mathematics in 1989, the National Council of Teachers of Mathematics (NCTM) spawned a major reform movement in school mathematics. The movement calls for abandoning curricula that promote thinking about
mathematics as a rigid system of externally dictated rules governed by standards of accuracy, speed, and memory. . . . A mathematics curriculum that emphasizes computation and rules is like a writing curriculum that emphasizes grammar and spelling; both put the cart before the horse. There is no place in a proper curriculum for mindless mimicry mathematics.[1]
Instead, proponents of reform envision classrooms in which students
have numerous and various interrelated experiences which allow them to solve complex problems; to read, write, and discuss mathematics; to conjecture, test, and build arguments about a conjecture's validity; to value the mathematical enterprise, the mathematical habits of mind, and the role of mathematics in human affairs; and to be encouraged to explore, guess, and even make errors so that they gain confidence in their own actions.[2]
To appreciate the magnitude of such reform, it is important to recognize that fundamental change is being called for in two areas. The first is the content of school mathematics. Historically, computational skill has been considered the most important part of mathematics for the masses.[3] To be sure, mathematics educators of the past lamented the fact that students often did not understand concepts or why certain procedures worked. But there seemed to be universal agreement that computation was important. Computational topics drove the mathematics curriculum, especially in the elementary years.
However, the situation has changed dramatically over the last decade. Technological advances have all but eliminated the need for paper-and-pencil computational skill. As a result, a major thrust of the reform movement has been the effort to replace the current obsolete, mathematics-as-computation curriculum with a mathematics curriculum that genuinely embraces conceptual understanding, reasoning, and problem solving as the fundamental goals of instruction.
The second area in which fundamental change is being sought is in the way we view teaching and learning. When computation dominated the mathematics curriculum, the prevailing psychological view of mathematics learning was behaviorism, and attention was focused on observable behaviors, not on mathematical thinking. Education (generating understanding) and training (producing specific performance) were confused.[4] Views of school mathematics and school learning were thus mutually reinforcing: school mathematics was seen as a set of computational skills; mathematics learning was seen as progressing through carefully scripted schedules of skill acquisition.
But current research in learning has uncovered deficiencies in instructional approaches based on behaviorism. According to the National Research Council:
Research in learning shows that students actually construct their own understanding based on new experiences that enlarge the intellectual framework in which ideas can be created. . . . Much of the failure in school mathematics is due to a tradition of teaching that is inappropriate to the way most students learn.[5]
Thus we are at a unique crossroads in American mathematics education. At a time when the quantitative literacy of our citizens is recognized as essential to our nation's future, we find that the mathematics we are teaching our children and the methods we are using to teach it are obsolete and ineffective. Unfortunately, the prevailing view of educators and the public at large is that mathematics consists of set procedures and that teaching means telling students how to perform those procedures. MATHEMATICS AS SET PROCEDURES
Mary is an experienced elementary teacher who is concerned about her students' learning of mathematics. She recognizes that the current curriculum does not interest her students and promotes failure among them. Because she actively seeks out interesting instructional activities even if they seem to have little to do with the textbook, Mary asked me for some suggestions on "something different and interesting" to do in her sixth-grade mathematics class.
The theme of the activities I suggested was baseball, and one of the activities asked students to examine the concept of a batter's "slugging average." In the official rules of baseball, this statistic is defined as the total bases achieved on all safe hits, divided by the total number of times at bat. In small groups, students were to discuss and analyze the problem until they constructed an understanding of the concept of slugging average and were able to use this understanding to devise a way of computing it. The goal of the activity was for students to make sense of a real-world use of mathematics, to get them involved in "problem formulation, problem solving, and mathematical reasoning."[6]
As Mary examined the activity, she became frustrated. "The [student] sheet does not tell you how to calculate the slugging average. How are you supposed to find it? If I can't figure it out, how will my students be able to do it?" As we talked about the activity, I tried to show how Mary might help her students make sense of the task as a problem-solving activity. She, however, focused on finding a step-by-step computational rule that she could teach her students: "It is important that I know exactly how to do it. I need to be able to show my students."
For Mary, understanding the mathematical idea of slugging average seemed to mean reducing it to a step-by-step, computational procedure. Her students would be learning mathematics if they learned to perform the procedure. This belief caused her to be frustrated with the instructional activity for two reasons. First, the activity required a substantial amount of thought and problem solving; the relevant computations were not immediately apparent. Because she did not treat the task as a problem of sense-making, as a puzzle to be solved, she viewed her inability to quickly find a solution as a failure to do mathematics properly. Second, because of her conception of the nature of mathematics, Mary did not understand why making sense of the problem should be part of the instructional activity. She believed that learning mathematics involved learning set procedures, that the procedures should be spelled out in detail, and that there should be no puzzlement either for her or for her students.
Jack was a novice elementary teacher with a special six-course concentration in mathematics. I observed him teach a statistics lesson on measures of central tendency. As presented in his textbook, the lesson was designed for students to calculate the mean, median, and mode from a set of data displayed in a frequency chart -- not a straightforward task. Jack attempted to show his students a clear set of procedures for calculating the statistics. But most of them could not even identify the individual data points. Thus the only way the students could use the procedures Jack was telling them was to follow them by rote.
The students became confused and grew frustrated. Even when Jack asked worthwhile questions that might promote conceptual understanding, he focused only on correct procedures. If a student's answer was not consistent with his goal of getting correct procedures, Jack told the student that it was wrong. He did not listen carefully to his students' responses; instead he evaluated them for consistency with his procedural goals. He did not try to make sense of his students' ideas so that he could guide them from their present thinking to more sophisticated thought. He simply attempted to coach his students to perform the procedures given in the text.
As we talked about the lesson afterward, Jack expressed frustration. He cared deeply about his students' learning of mathematics, but he did not understand why they were having difficulty. He blamed himself for not teaching well enough and the students for not trying hard enough. He told me that he thought that, to learn mathematics, students must learn the procedures: "That's the way I learned it; that was effective for me." Because Jack conceived of mathematics as following set procedures, he was unable to see that his focus on procedures made it impossible for his students to make sense out of mathematics. In fact, because of his restricted view of mathematics, he was unable to understand the causes of his students' learning difficulties.
A study by Paul Cobb and his colleagues illustrates how some teachers extend the conception of mathematics-as-procedures even to topics that are not computational in nature.[7] A third-grade teacher was teaching the concept of place value. All her instructional moves suggested that she believed that students "understand" mathematics when they can successfully follow procedural instructions. All her instructional discourse was interpreted (by her and by her students) as instructions to be followed. For example, in one instructional activity the students looked at pictures of tens and ones blocks and answered a series of questions -- "How many tens? How many ones? What number is that?" --with a memorized litany of responses. (There was never any discussion about why these responses were valid.) Cobb and his colleagues concluded that, for students to be successful in this class, they did not need to create and operate on meaningful mental constructs. They merely had to learn to follow the teacher's instructions. Furthermore, the researchers reported that there were almost no opportunities for explanation or justification in this classroom, though these are essential activities for members of a community trying to make sense of mathematics.
All three of these teachers unintentionally taught what the National Research Council has termed "mindless mimicry mathematics." They did so, not because they were unconcerned about their students' learning, but because of their own mistaken beliefs about the nature of mathematics. Indeed, because they wanted to ensure that their students would be successful in completing the "mathematical" tasks given to them, they attempted to reduce these tasks to rigid, step-by-step procedures. They reasoned that, if the students followed these steps, they could not fail.
But by reducing mathematics to the following of set procedures, these teachers were inadvertently robbing their students of opportunities to "do" mathematics. Because students' intuitive ideas about making sense of mathematics were ignored, and therefore devalued, the development of their mathematical reasoning skills was impeded. The result of this type of instruction is that students' personal mathematical ideas become totally disconnected from the formal mathematics they learn in school. As James Hiebert comments about students in a traditional curriculum:
A significant difference exists between the way in which children solve problems outside of school and inside of school. In nonschool settings, children seem to interpret the problems appropriately, use strategies they understand to solve problems, and judge the reasonableness of the answers. In contrast, many children are unable to interpret school tasks in a meaningful way, frequently apply memorized but little-understood rules to solve them, and have little idea of whether their answers make sense. Outside of school, many children seem to use their intuitions and conceptual understandings to decide what to do, what strategy to use. Inside of school, many children try to recall and execute rules for solving problems "like this one" to find the answer. [8] TEACHER BELIEFS AND REFORM
What makes the episodes described above even more alarming is that these teachers' beliefs not only caused them to implement an inappropriate curriculum but also blocked their understanding and acceptance of the philosophy of the reform movement, thereby precluding the possibility of substantive curricular change. These teachers possessed a view of mathematics that is totally incongruous with that of the current reform movement.
While proponents of reform see students' struggle with problems such as determining a player's slugging average as "doing" mathematics, those who view mathematics as the learning of rules view such struggle as an instructional deficiency. While reformers view such activities to be successful if students persevere and eventually succeed in their sense-making efforts, those who believe in mathematics as rule-learning judge students to be successful if they learn to compute slugging averages quickly and easily. Those who conceptualize mathematics as rule-learning view teaching as clearly telling students how to perform set procedures. Reformers view mathematics as thinking and reasoning; they view teaching as involving and guiding students in the process of making sense of mathematical ideas. Indeed, as the chairperson of the commission that wrote the NCTM Standards stated, "The single most compelling issue in improving school mathematics is to change the epistemology of mathematics in schools, the sense on the part of teachers and students of what the mathematical enterprise is all about."[9]
Although a change in epistemology is necessary for the reform of school mathematics, it is not sufficient. Even when teachers begin to understand and become willing to implement reform, they must acquire knowledge and competencies that their traditional beliefs have heretofore prevented them from acquiring. That is, teachers who have always believed that mathematics consists of following set procedures invented by others will have little experience making sense out of mathematics. It will be difficult for them to understand how to guide and support students' invention of mathematical ideas. It will be difficult for them to know how to create an intellectual and social climate in the classroom so that students discuss, reflect on, and make sense of tasks rather than merely learn to perform mindless symbolic mimicry. Even if these teachers are provided with good instructional materials and instructional scripts for exploring ideas with students, what will they do if students do not follow a script? Will they be able to improvise in a way that is consistent with the goals of exploration and sense-making? How likely is it that they will be able to "respond constructively to unexpected conjectures that emerge as students follow their own paths in approaching mathematical problems"?[10]
Furthermore, because its instructional goals are cognitive rather than behavioral and because it seeks to mold students' own personal mathematical ideas, teaching that is consistent with the reform movement requires an extensive knowledge of how students learn mathematics. Teaching based on a "constructivist" view of learning must be guided by knowledge of the conceptual advances that students need to make for various mathematical topics and of the processes by which they make these advances. As Leslie Steffe and Paul Cobb note:
From our perspective, the essential pedagogical task is not to instill "correct ways of doing" but rather to guide children's constructive activities until they eventually "find" viable techniques. Such guidance must necessarily start from points that are accessible to the children. In order to establish these starting points, we must first gain insight into the children's conceptual structures and methods, no matter how wayward or ineffective they might seem to us. The teacher will be far more successful in accommodating children's growth in conceptual understanding if he or she has some notion of what the child's present structures and ways of operating are.[11]
However, teachers who are accustomed to implementing the traditional, procedural/behavioral curriculum have not needed much knowledge of how children learn mathematics. They have been required only to explain to students set sequences of procedures prescribed by textbooks. Thus teachers accustomed to teaching the traditional curriculum may lack knowledge about mathematics and student learning that is essential to implementing a reform curriculum. REASONS FOR TEACHERS' BELIEFS
Although my above comments have been restricted to the elementary level, there is a similar emphasis on procedural skills at both the secondary and university levels.[12] But why are teachers' beliefs so incongruous with those of the current reform movement? Should we blame teachers because they ought to know better? Nothing could be further from the truth. Like most adults, almost all current teachers were educated at the elementary, secondary, and university levels in curricula that promoted the conception of mathematics as procedures rather than as sense-making. Moreover, the school environments in which teachers now teach demand this rule-based view of mathematics. Their mathematics textbooks support it. State and district testing programs assess adherence to it. Most parents, school officials, and politicians -- all of whom dictate curricula to teachers -- also see mathematics as sets of rules to follow.
Unfortunately, today's teachers have been caught in the midst of a paradigm shift. Accepted views about what and how mathematics should be taught have changed drastically since most teachers were in school. In fact, one of the most serious obstacles to reform is that the current mathematics curriculum is self-perpetuating. Teachers who are asked to teach the reformed mathematics curricula are products of an old curriculum that developed in them beliefs so incompatible with those of the new curricula that they can understand many of the innovations only with great effort. We are caught in a pernicious cycle of mathematical mislearning. PATHS TO REFORM
What can be done? I will discuss two possible ways that the education system can help teachers deal with the reform movement. One solution focuses on universities; the other, on school districts.
University teacher education. One oft-proposed solution focuses on elementary teachers, but the discussion has implications for secondary teachers as well. It is argued that U.S. elementary teachers -- who are typically required to take only one "methods of teaching" course and two mathematics courses -- simply have not studied enough mathematics to develop the belief system and conceptual foundation necessary to teach mathematics in the ways suggested by the reform movement. In one variant of this proposal, additional mathematics course work would be required of all prospective elementary teachers. In another variant, suggested by the National Research Council, a mathematics specialization would be required.
The United States is one of the few countries in the world that continues to pretend -- despite substantial evidence to the contrary -- that elementary school teachers are able to teach all subjects equally well. It is time that we identify a cadre of teachers with special interests in mathematics and science who would be well prepared to teach young children both mathematics and science in an integrated, discovery-based environment.[13]
According to the latter proposal, every elementary teacher would choose one or two areas of specialization. Teaching elementary school mathematics would require not only a special interest in mathematics, but extensive training in mathematics and mathematical pedagogy. Pre-service elementary teachers who specialize in mathematics might take, for instance, five or six mathematics courses and one or two mathematics methods courses. Current elementary teachers who wish to continue teaching mathematics would be required to take additional courses in mathematics and mathematics education.
However, if this proposal is to solve the problem, the additional mathematics that teachers take must be taught properly. That is, it must be taught as sense-making. Unfortunately, most university mathematics courses reinforce rather than debunk the view of mathematics as a set of procedures to be memorized. They present mathematics "only in the authoritarian framework of Moses coming down from Mt. Sinai."[14] Because such courses simply perpetuate the mathematical miseducation that occurs in grades K-12, requiring teachers to take more of them will do little to solve the problem. Recall Jack, who had taken six university mathematics courses and had done well in each. Moreover, secondary teachers and university mathematics professors, who have extensive backgrounds in mathematics, have shown themselves to be just as prone as elementary teachers to teach mathematics as procedures.
To help solve the problem rather than exacerbate it, teacher education institutions need to offer numerous mathematics courses for teachers that treat mathematics as sense-making, not rule-following. While well-taught "methods of teaching" courses can help alleviate the problem by attempting to teach mathematics and mathematical epistemology in addition to pedagogy, it is not likely that one or two courses alone can undo the damage of more than 13 years of mathematical miseducation. As the National Research Council states, "Teachers themselves need experience in doing mathematics -- in exploring, guessing, testing, estimating, arguing, and proving. . . . [They] should learn mathematics in a manner that encourages active engagement with mathematical ideas."[15] Teachers must be taught mathematics properly before we can expect them to teach it properly. Universities must take the lead in making changes in the way mathematics is taught.
School district in service training. The second proposed solution lies with in service training designed to help practicing teachers better understand mathematics and learn to implement the recommendations of the reform movement. Typically, school districts offer one- or two-day workshops to provide teachers with ideas and materials for innovative instructional activities. However, these "make-and-take workshops" fail to cause the changes called for by the NCTM Standards because they do not address teachers' underlying pedagogical philosophies, their knowledge and beliefs about mathematics, or their knowledge of the processes by which students come to understand mathematical ideas.
Indeed, as we can see from the activity regarding Slugging averages, if a teacher's beliefs are not consistent with those of an instructional activity, he or she will be unable to understand or achieve the goals of the activity, no matter how well-written it is or how much guidance is given in "teacher notes." Moreover, even if teachers properly implement activities given to them, sprinkling five or 10 innovative instructional activities into the standard rule-based curriculum will not greatly affect students' learning. It will only give teachers the illusion that substantive changes are taking place in their mathematics classes.
More promising than isolated instructional activities are extensive programs of in service training that not only provide comprehensive sets of curriculum materials but also offer instruction in mathematics and mathematics learning. The curriculum materials provided must be extensive enough to allow teachers to implement a new approach to mathematics every day -- or at least for several prolonged instructional periods. The in service programs must last at least several weeks, must continue throughout the year, and must demand that teachers learn both pedagogy and mathematics. Only by learning mathematics properly can teachers become convinced that mathematics consists of problem solving and sense-making and gain skill in using the processes and strategies required for mathematical sense-making. Moreover, with a successful program of in service training, teachers' learning will not end with the completion of the program; it will continue every day in their classrooms. Watching their students' sense-making efforts will cause teachers to continually refine their own conceptions of mathematical ideas and of mathematics learning.
Of course, such high-quality in service programs are expensive. Without them, however, it is highly unlikely that teachers will be willing or able to make the changes called for by the reform movement. THE CHALLENGE AHEAD
All our efforts to make the mathematics curriculum consistent with the NCTM Standards will fail if teachers' beliefs about mathematics do not become aligned with those of the reform movement. For teacher educators, school officials, political leaders, and teachers themselves to acknowledge that there is a serious problem with the way our society views mathematics is the first step.
The next step requires massive reform on the part of all the institutions that affect the education and working environment of teachers. It is unfair -- and unproductive -- merely to demand that teachers see and teach mathematics in a different way. Through extensive education programs and institutional reform, we must help teachers become comfortable with this new view of mathematics. And, because they are dedicated professionals, once they understand and believe in the reform movement, teachers will lead the way in implementing it.
National Research Council, Everybody Counts: A Report to the Nation on the Future of Mathematics Education (Washington, D.C.: National Academy Press, 1989), p. 44.
Curriculum and Evaluation Standards for School Mathematics (Reston, Va.: National Council of Teachers of Mathematics, 1989), p. 12.
Lauren B. Resnick, Education and Learning to Think (Washington, D.C.: National Academy Press, 1987).
Ernst von Glaserfeld, Introduction to idem, ed., Radical Constructivism in Mathematics Education (Dordrecht: Kiuwer Academic Publishers, 1991), pp. xiii-xix.
National Research Council, p. 6.
Curriculum and Evaluation Standards, p. 25.
Paul Cobb et al., "Characteristics of Classroom Mathematics Traditions: An Interactional Analysis," American Educational Research Journal, vol. 29, 1992, pp. 573-604.
James Hiebert, "The Struggle to Link Written Symbols with Understandings: An Update," Arithmetic Teacher, March 1989, p. 39.
Thomas A. Romberg, "Further Thoughts on the Standards: A Reaction to Apple," Journal of Research in Mathematics Education, vol. 23, 1992, p. 433.
National Research Council, p. 65.
Leslie P. Steffe and Paul Cobb, Construction of Arithmetical Meanings and Strategies (New York: Springer-Verlag, 1988), pp. vii-viii.
Deborah Lowenberg Ball, 'Prospective Elementary Teachers' Understanding of Division," Journal of Research in Mathematics Education, vol. 21, 1990, pp. 132-44; and National Research Council, op. cit.
National Research Council, p. 64.
Ibid., p. 66.
Ibid., pp. 65-66.
ILLUSTRATION: It only takes a simple swing to make things implemented.
CARTOON: Write the largest number you can.
CARTOON: Are you sure it won't grow up to be a math book, - rodaniel Oct 23, 2009- rodaniel Oct 23, 2009
By MICHAEL T. BATTISTA
MICHAEL T. BATTISTA is a professor of mathematics education at Kent State University, Kent, Ohio. He is a former member of the editorial panel of the Journal of Research in Mathematics Education and has written extensively on the learning and teaching of mathematics.
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TEACHER BELIEFS AND THE REFORM MOVEMENT IN MATHEMATICS EDUCATION
Contents
- THE REFORM MOVEMENT
- MATHEMATICS AS SET PROCEDURES
- TEACHER BELIEFS AND REFORM
- REASONS FOR TEACHERS' BELIEFS
- PATHS TO REFORM
- THE CHALLENGE AHEAD
Through extensive education programs and institutional reform, we must help teachers become comfortable with the new view of mathematics, Mr. Battista says -- because, once they fully understand and believe in the reform movement, teachers will lead the way in implementing it.Teachers are key to the success of the current reform movement in U.S. mathematics education. However, many teachers have beliefs about mathematics that are incompatible with those underlying the reform effort. Because these beliefs play a critical role not only in what teachers teach but in how they teach it, this incompatibility blocks reform and prolongs the use of a mathematics curriculum that is seriously damaging the mathematical health of our children.
THE REFORM MOVEMENT
With its release of Curriculum and Evaluation Standards for School Mathematics in 1989, the National Council of Teachers of Mathematics (NCTM) spawned a major reform movement in school mathematics. The movement calls for abandoning curricula that promote thinking about
mathematics as a rigid system of externally dictated rules governed by standards of accuracy, speed, and memory. . . . A mathematics curriculum that emphasizes computation and rules is like a writing curriculum that emphasizes grammar and spelling; both put the cart before the horse. There is no place in a proper curriculum for mindless mimicry mathematics.[1]
Instead, proponents of reform envision classrooms in which students
have numerous and various interrelated experiences which allow them to solve complex problems; to read, write, and discuss mathematics; to conjecture, test, and build arguments about a conjecture's validity; to value the mathematical enterprise, the mathematical habits of mind, and the role of mathematics in human affairs; and to be encouraged to explore, guess, and even make errors so that they gain confidence in their own actions.[2]
To appreciate the magnitude of such reform, it is important to recognize that fundamental change is being called for in two areas. The first is the content of school mathematics. Historically, computational skill has been considered the most important part of mathematics for the masses.[3] To be sure, mathematics educators of the past lamented the fact that students often did not understand concepts or why certain procedures worked. But there seemed to be universal agreement that computation was important. Computational topics drove the mathematics curriculum, especially in the elementary years.
However, the situation has changed dramatically over the last decade. Technological advances have all but eliminated the need for paper-and-pencil computational skill. As a result, a major thrust of the reform movement has been the effort to replace the current obsolete, mathematics-as-computation curriculum with a mathematics curriculum that genuinely embraces conceptual understanding, reasoning, and problem solving as the fundamental goals of instruction.
The second area in which fundamental change is being sought is in the way we view teaching and learning. When computation dominated the mathematics curriculum, the prevailing psychological view of mathematics learning was behaviorism, and attention was focused on observable behaviors, not on mathematical thinking. Education (generating understanding) and training (producing specific performance) were confused.[4] Views of school mathematics and school learning were thus mutually reinforcing: school mathematics was seen as a set of computational skills; mathematics learning was seen as progressing through carefully scripted schedules of skill acquisition.
But current research in learning has uncovered deficiencies in instructional approaches based on behaviorism. According to the National Research Council:
Research in learning shows that students actually construct their own understanding based on new experiences that enlarge the intellectual framework in which ideas can be created. . . . Much of the failure in school mathematics is due to a tradition of teaching that is inappropriate to the way most students learn.[5]
Thus we are at a unique crossroads in American mathematics education. At a time when the quantitative literacy of our citizens is recognized as essential to our nation's future, we find that the mathematics we are teaching our children and the methods we are using to teach it are obsolete and ineffective. Unfortunately, the prevailing view of educators and the public at large is that mathematics consists of set procedures and that teaching means telling students how to perform those procedures.
MATHEMATICS AS SET PROCEDURES
Mary is an experienced elementary teacher who is concerned about her students' learning of mathematics. She recognizes that the current curriculum does not interest her students and promotes failure among them. Because she actively seeks out interesting instructional activities even if they seem to have little to do with the textbook, Mary asked me for some suggestions on "something different and interesting" to do in her sixth-grade mathematics class.
The theme of the activities I suggested was baseball, and one of the activities asked students to examine the concept of a batter's "slugging average." In the official rules of baseball, this statistic is defined as the total bases achieved on all safe hits, divided by the total number of times at bat. In small groups, students were to discuss and analyze the problem until they constructed an understanding of the concept of slugging average and were able to use this understanding to devise a way of computing it. The goal of the activity was for students to make sense of a real-world use of mathematics, to get them involved in "problem formulation, problem solving, and mathematical reasoning."[6]
As Mary examined the activity, she became frustrated. "The [student] sheet does not tell you how to calculate the slugging average. How are you supposed to find it? If I can't figure it out, how will my students be able to do it?" As we talked about the activity, I tried to show how Mary might help her students make sense of the task as a problem-solving activity. She, however, focused on finding a step-by-step computational rule that she could teach her students: "It is important that I know exactly how to do it. I need to be able to show my students."
For Mary, understanding the mathematical idea of slugging average seemed to mean reducing it to a step-by-step, computational procedure. Her students would be learning mathematics if they learned to perform the procedure. This belief caused her to be frustrated with the instructional activity for two reasons. First, the activity required a substantial amount of thought and problem solving; the relevant computations were not immediately apparent. Because she did not treat the task as a problem of sense-making, as a puzzle to be solved, she viewed her inability to quickly find a solution as a failure to do mathematics properly. Second, because of her conception of the nature of mathematics, Mary did not understand why making sense of the problem should be part of the instructional activity. She believed that learning mathematics involved learning set procedures, that the procedures should be spelled out in detail, and that there should be no puzzlement either for her or for her students.
Jack was a novice elementary teacher with a special six-course concentration in mathematics. I observed him teach a statistics lesson on measures of central tendency. As presented in his textbook, the lesson was designed for students to calculate the mean, median, and mode from a set of data displayed in a frequency chart -- not a straightforward task. Jack attempted to show his students a clear set of procedures for calculating the statistics. But most of them could not even identify the individual data points. Thus the only way the students could use the procedures Jack was telling them was to follow them by rote.
The students became confused and grew frustrated. Even when Jack asked worthwhile questions that might promote conceptual understanding, he focused only on correct procedures. If a student's answer was not consistent with his goal of getting correct procedures, Jack told the student that it was wrong. He did not listen carefully to his students' responses; instead he evaluated them for consistency with his procedural goals. He did not try to make sense of his students' ideas so that he could guide them from their present thinking to more sophisticated thought. He simply attempted to coach his students to perform the procedures given in the text.
As we talked about the lesson afterward, Jack expressed frustration. He cared deeply about his students' learning of mathematics, but he did not understand why they were having difficulty. He blamed himself for not teaching well enough and the students for not trying hard enough. He told me that he thought that, to learn mathematics, students must learn the procedures: "That's the way I learned it; that was effective for me." Because Jack conceived of mathematics as following set procedures, he was unable to see that his focus on procedures made it impossible for his students to make sense out of mathematics. In fact, because of his restricted view of mathematics, he was unable to understand the causes of his students' learning difficulties.
A study by Paul Cobb and his colleagues illustrates how some teachers extend the conception of mathematics-as-procedures even to topics that are not computational in nature.[7] A third-grade teacher was teaching the concept of place value. All her instructional moves suggested that she believed that students "understand" mathematics when they can successfully follow procedural instructions. All her instructional discourse was interpreted (by her and by her students) as instructions to be followed. For example, in one instructional activity the students looked at pictures of tens and ones blocks and answered a series of questions -- "How many tens? How many ones? What number is that?" --with a memorized litany of responses. (There was never any discussion about why these responses were valid.) Cobb and his colleagues concluded that, for students to be successful in this class, they did not need to create and operate on meaningful mental constructs. They merely had to learn to follow the teacher's instructions. Furthermore, the researchers reported that there were almost no opportunities for explanation or justification in this classroom, though these are essential activities for members of a community trying to make sense of mathematics.
All three of these teachers unintentionally taught what the National Research Council has termed "mindless mimicry mathematics." They did so, not because they were unconcerned about their students' learning, but because of their own mistaken beliefs about the nature of mathematics. Indeed, because they wanted to ensure that their students would be successful in completing the "mathematical" tasks given to them, they attempted to reduce these tasks to rigid, step-by-step procedures. They reasoned that, if the students followed these steps, they could not fail.
But by reducing mathematics to the following of set procedures, these teachers were inadvertently robbing their students of opportunities to "do" mathematics. Because students' intuitive ideas about making sense of mathematics were ignored, and therefore devalued, the development of their mathematical reasoning skills was impeded. The result of this type of instruction is that students' personal mathematical ideas become totally disconnected from the formal mathematics they learn in school. As James Hiebert comments about students in a traditional curriculum:
A significant difference exists between the way in which children solve problems outside of school and inside of school. In nonschool settings, children seem to interpret the problems appropriately, use strategies they understand to solve problems, and judge the reasonableness of the answers. In contrast, many children are unable to interpret school tasks in a meaningful way, frequently apply memorized but little-understood rules to solve them, and have little idea of whether their answers make sense. Outside of school, many children seem to use their intuitions and conceptual understandings to decide what to do, what strategy to use. Inside of school, many children try to recall and execute rules for solving problems "like this one" to find the answer. [8]
TEACHER BELIEFS AND REFORM
What makes the episodes described above even more alarming is that these teachers' beliefs not only caused them to implement an inappropriate curriculum but also blocked their understanding and acceptance of the philosophy of the reform movement, thereby precluding the possibility of substantive curricular change. These teachers possessed a view of mathematics that is totally incongruous with that of the current reform movement.
While proponents of reform see students' struggle with problems such as determining a player's slugging average as "doing" mathematics, those who view mathematics as the learning of rules view such struggle as an instructional deficiency. While reformers view such activities to be successful if students persevere and eventually succeed in their sense-making efforts, those who believe in mathematics as rule-learning judge students to be successful if they learn to compute slugging averages quickly and easily. Those who conceptualize mathematics as rule-learning view teaching as clearly telling students how to perform set procedures. Reformers view mathematics as thinking and reasoning; they view teaching as involving and guiding students in the process of making sense of mathematical ideas. Indeed, as the chairperson of the commission that wrote the NCTM Standards stated, "The single most compelling issue in improving school mathematics is to change the epistemology of mathematics in schools, the sense on the part of teachers and students of what the mathematical enterprise is all about."[9]
Although a change in epistemology is necessary for the reform of school mathematics, it is not sufficient. Even when teachers begin to understand and become willing to implement reform, they must acquire knowledge and competencies that their traditional beliefs have heretofore prevented them from acquiring. That is, teachers who have always believed that mathematics consists of following set procedures invented by others will have little experience making sense out of mathematics. It will be difficult for them to understand how to guide and support students' invention of mathematical ideas. It will be difficult for them to know how to create an intellectual and social climate in the classroom so that students discuss, reflect on, and make sense of tasks rather than merely learn to perform mindless symbolic mimicry. Even if these teachers are provided with good instructional materials and instructional scripts for exploring ideas with students, what will they do if students do not follow a script? Will they be able to improvise in a way that is consistent with the goals of exploration and sense-making? How likely is it that they will be able to "respond constructively to unexpected conjectures that emerge as students follow their own paths in approaching mathematical problems"?[10]
Furthermore, because its instructional goals are cognitive rather than behavioral and because it seeks to mold students' own personal mathematical ideas, teaching that is consistent with the reform movement requires an extensive knowledge of how students learn mathematics. Teaching based on a "constructivist" view of learning must be guided by knowledge of the conceptual advances that students need to make for various mathematical topics and of the processes by which they make these advances. As Leslie Steffe and Paul Cobb note:
From our perspective, the essential pedagogical task is not to instill "correct ways of doing" but rather to guide children's constructive activities until they eventually "find" viable techniques. Such guidance must necessarily start from points that are accessible to the children. In order to establish these starting points, we must first gain insight into the children's conceptual structures and methods, no matter how wayward or ineffective they might seem to us. The teacher will be far more successful in accommodating children's growth in conceptual understanding if he or she has some notion of what the child's present structures and ways of operating are.[11]
However, teachers who are accustomed to implementing the traditional, procedural/behavioral curriculum have not needed much knowledge of how children learn mathematics. They have been required only to explain to students set sequences of procedures prescribed by textbooks. Thus teachers accustomed to teaching the traditional curriculum may lack knowledge about mathematics and student learning that is essential to implementing a reform curriculum.
REASONS FOR TEACHERS' BELIEFS
Although my above comments have been restricted to the elementary level, there is a similar emphasis on procedural skills at both the secondary and university levels.[12] But why are teachers' beliefs so incongruous with those of the current reform movement? Should we blame teachers because they ought to know better? Nothing could be further from the truth. Like most adults, almost all current teachers were educated at the elementary, secondary, and university levels in curricula that promoted the conception of mathematics as procedures rather than as sense-making. Moreover, the school environments in which teachers now teach demand this rule-based view of mathematics. Their mathematics textbooks support it. State and district testing programs assess adherence to it. Most parents, school officials, and politicians -- all of whom dictate curricula to teachers -- also see mathematics as sets of rules to follow.
Unfortunately, today's teachers have been caught in the midst of a paradigm shift. Accepted views about what and how mathematics should be taught have changed drastically since most teachers were in school. In fact, one of the most serious obstacles to reform is that the current mathematics curriculum is self-perpetuating. Teachers who are asked to teach the reformed mathematics curricula are products of an old curriculum that developed in them beliefs so incompatible with those of the new curricula that they can understand many of the innovations only with great effort. We are caught in a pernicious cycle of mathematical mislearning.
PATHS TO REFORM
What can be done? I will discuss two possible ways that the education system can help teachers deal with the reform movement. One solution focuses on universities; the other, on school districts.
University teacher education. One oft-proposed solution focuses on elementary teachers, but the discussion has implications for secondary teachers as well. It is argued that U.S. elementary teachers -- who are typically required to take only one "methods of teaching" course and two mathematics courses -- simply have not studied enough mathematics to develop the belief system and conceptual foundation necessary to teach mathematics in the ways suggested by the reform movement. In one variant of this proposal, additional mathematics course work would be required of all prospective elementary teachers. In another variant, suggested by the National Research Council, a mathematics specialization would be required.
The United States is one of the few countries in the world that continues to pretend -- despite substantial evidence to the contrary -- that elementary school teachers are able to teach all subjects equally well. It is time that we identify a cadre of teachers with special interests in mathematics and science who would be well prepared to teach young children both mathematics and science in an integrated, discovery-based environment.[13]
According to the latter proposal, every elementary teacher would choose one or two areas of specialization. Teaching elementary school mathematics would require not only a special interest in mathematics, but extensive training in mathematics and mathematical pedagogy. Pre-service elementary teachers who specialize in mathematics might take, for instance, five or six mathematics courses and one or two mathematics methods courses. Current elementary teachers who wish to continue teaching mathematics would be required to take additional courses in mathematics and mathematics education.
However, if this proposal is to solve the problem, the additional mathematics that teachers take must be taught properly. That is, it must be taught as sense-making. Unfortunately, most university mathematics courses reinforce rather than debunk the view of mathematics as a set of procedures to be memorized. They present mathematics "only in the authoritarian framework of Moses coming down from Mt. Sinai."[14] Because such courses simply perpetuate the mathematical miseducation that occurs in grades K-12, requiring teachers to take more of them will do little to solve the problem. Recall Jack, who had taken six university mathematics courses and had done well in each. Moreover, secondary teachers and university mathematics professors, who have extensive backgrounds in mathematics, have shown themselves to be just as prone as elementary teachers to teach mathematics as procedures.
To help solve the problem rather than exacerbate it, teacher education institutions need to offer numerous mathematics courses for teachers that treat mathematics as sense-making, not rule-following. While well-taught "methods of teaching" courses can help alleviate the problem by attempting to teach mathematics and mathematical epistemology in addition to pedagogy, it is not likely that one or two courses alone can undo the damage of more than 13 years of mathematical miseducation. As the National Research Council states, "Teachers themselves need experience in doing mathematics -- in exploring, guessing, testing, estimating, arguing, and proving. . . . [They] should learn mathematics in a manner that encourages active engagement with mathematical ideas."[15] Teachers must be taught mathematics properly before we can expect them to teach it properly. Universities must take the lead in making changes in the way mathematics is taught.
School district in service training. The second proposed solution lies with in service training designed to help practicing teachers better understand mathematics and learn to implement the recommendations of the reform movement. Typically, school districts offer one- or two-day workshops to provide teachers with ideas and materials for innovative instructional activities. However, these "make-and-take workshops" fail to cause the changes called for by the NCTM Standards because they do not address teachers' underlying pedagogical philosophies, their knowledge and beliefs about mathematics, or their knowledge of the processes by which students come to understand mathematical ideas.
Indeed, as we can see from the activity regarding Slugging averages, if a teacher's beliefs are not consistent with those of an instructional activity, he or she will be unable to understand or achieve the goals of the activity, no matter how well-written it is or how much guidance is given in "teacher notes." Moreover, even if teachers properly implement activities given to them, sprinkling five or 10 innovative instructional activities into the standard rule-based curriculum will not greatly affect students' learning. It will only give teachers the illusion that substantive changes are taking place in their mathematics classes.
More promising than isolated instructional activities are extensive programs of in service training that not only provide comprehensive sets of curriculum materials but also offer instruction in mathematics and mathematics learning. The curriculum materials provided must be extensive enough to allow teachers to implement a new approach to mathematics every day -- or at least for several prolonged instructional periods. The in service programs must last at least several weeks, must continue throughout the year, and must demand that teachers learn both pedagogy and mathematics. Only by learning mathematics properly can teachers become convinced that mathematics consists of problem solving and sense-making and gain skill in using the processes and strategies required for mathematical sense-making. Moreover, with a successful program of in service training, teachers' learning will not end with the completion of the program; it will continue every day in their classrooms. Watching their students' sense-making efforts will cause teachers to continually refine their own conceptions of mathematical ideas and of mathematics learning.
Of course, such high-quality in service programs are expensive. Without them, however, it is highly unlikely that teachers will be willing or able to make the changes called for by the reform movement.
THE CHALLENGE AHEAD
All our efforts to make the mathematics curriculum consistent with the NCTM Standards will fail if teachers' beliefs about mathematics do not become aligned with those of the reform movement. For teacher educators, school officials, political leaders, and teachers themselves to acknowledge that there is a serious problem with the way our society views mathematics is the first step.
The next step requires massive reform on the part of all the institutions that affect the education and working environment of teachers. It is unfair -- and unproductive -- merely to demand that teachers see and teach mathematics in a different way. Through extensive education programs and institutional reform, we must help teachers become comfortable with this new view of mathematics. And, because they are dedicated professionals, once they understand and believe in the reform movement, teachers will lead the way in implementing it.
- National Research Council, Everybody Counts: A Report to the Nation on the Future of Mathematics Education (Washington, D.C.: National Academy Press, 1989), p. 44.
- Curriculum and Evaluation Standards for School Mathematics (Reston, Va.: National Council of Teachers of Mathematics, 1989), p. 12.
- Lauren B. Resnick, Education and Learning to Think (Washington, D.C.: National Academy Press, 1987).
- Ernst von Glaserfeld, Introduction to idem, ed., Radical Constructivism in Mathematics Education (Dordrecht: Kiuwer Academic Publishers, 1991), pp. xiii-xix.
- National Research Council, p. 6.
- Curriculum and Evaluation Standards, p. 25.
- Paul Cobb et al., "Characteristics of Classroom Mathematics Traditions: An Interactional Analysis," American Educational Research Journal, vol. 29, 1992, pp. 573-604.
- James Hiebert, "The Struggle to Link Written Symbols with Understandings: An Update," Arithmetic Teacher, March 1989, p. 39.
- Thomas A. Romberg, "Further Thoughts on the Standards: A Reaction to Apple," Journal of Research in Mathematics Education, vol. 23, 1992, p. 433.
- National Research Council, p. 65.
- Leslie P. Steffe and Paul Cobb, Construction of Arithmetical Meanings and Strategies (New York: Springer-Verlag, 1988), pp. vii-viii.
- Deborah Lowenberg Ball, 'Prospective Elementary Teachers' Understanding of Division," Journal of Research in Mathematics Education, vol. 21, 1990, pp. 132-44; and National Research Council, op. cit.
- National Research Council, p. 64.
- Ibid., p. 66.
- Ibid., pp. 65-66.
ILLUSTRATION: It only takes a simple swing to make things implemented.CARTOON: Write the largest number you can.
CARTOON: Are you sure it won't grow up to be a math book,
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By MICHAEL T. BATTISTA
MICHAEL T. BATTISTA is a professor of mathematics education at Kent State University, Kent, Ohio. He is a former member of the editorial panel of the Journal of Research in Mathematics Education and has written extensively on the learning and teaching of mathematics.
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